Humans have the ability to locate a sound source with better than 5° accuracy in both azimuth and elevation. Humans also have the ability to perceive and approximate the distance of a source from them. In this regard, multiple cues may be used, including some that arise from sound scattering from the listener themselves (W. M. Hartmann, “How We Localize Sound”, Physics Today, November 1999, pp. 24-29).
The cues that arise due to scattering from the anatomy of the listener exhibit considerable person-to-person variability. These cues may be encapsulated in a transfer function that is termed the Head Realted Transfer Function (HRTF).
In order to recreate the sound pressure at the eardrums to make a synthetic audio scene indistinguishable from the real one, the virtual audio scene must include the HRTF-based cues to achieve accurate simulation (D. N. Zotkin, et al., “Creation of Virtual Auditory Spaces”, 2003, accepted IEEE Trans. Multimedia—available off authors' homepages).
The HRTF depends on the direction of arrival of the sound, and, for nearby sources, on the source distance. If the sound source is located at spherical coordinates (r, θ, φ), then the left and right HRTFs Hl and Hr are defined as the ratio of the complex sound pressure at the corresponding eardrum ψl,r to the free-field sound pressure at the center of the head ψf as if the listener is absent (R. O. Duda, et al., “Range Dependence of the Response of a Spherical Head Model”, J. Acoust. Soc. Am., 104, 1998, pp. 3048-3058).
                                          H                          l              ,              r                                ⁡                      (                          ω              ,              r              ,              θ              ,              φ                        )                          =                                            ψ                              l                ,                r                                      ⁡                          (                              ω                ,                r                ,                θ                ,                φ                            )                                                          ψ              f                        ⁡                          (              ω              )                                                          (        1        )            
To synthesize the audio scene given the source location (r,φ,θ) one needs to filter the signal with H(r,φ,θ) and the result rendered binaurally through headphones. To obtain the HRTFs for a given individual, an arrangement such as depicted in FIG. 1 is used. A source (speaker) is placed at a given location (r,θ,φ), and a generated sound is then recorded using a microphone placed in the ear canal of an individual. In order to obtain the HRTF corresponding to a different source location, the speaker is moved to that location and the measurement is repeated. The listener is required to remain stationary during this process in order that the location for the HRTF may be reliably described. HRTF measurements from thousands of points are needed, and the process is time-consuming, tedious and burdensome to the listener. One of the reasons spatial audio technology has been hampered is the unavailability of rapid HRTF measurement techniques.
Additionally, HRTF must be interpolated between discrete measurement positions to avoid audible jumps in sound. Many techniques have been proposed to perform the interpolation of the HRTF, however, proper interpolation is still regarded as an open question.
In addition, the dependence of the HRTF on the range r (distance between the source of the sound and the microphone) is also usually neglected since the HRTF measurements are tedious and time-consuming procedures. However, since the HRTF measured at a distance is known to be incorrect for relatively nearby sources, only relatively distant sources are simulated.
As a result of these inadequacies, HRTF measurement methods suffer from a lack of a complete range of measurements for the HRTF. However, many applications such as games, auditory user interfaces, entertainment, and virtual reality simulations demand the ability to accurately simulate sounds at relatively close ranges.
The Head Related Transfer Function characterizes the scattering properties of a person's anatomy (especially the pinnae, head and torso), and exhibits considerable person-to-person variability. Since the HRTF arises from a scattering process, it can be characterized as a solution of a scattering problem.
When a body with surface S scatters sound from a source located at (r1,θ1, φ1) the complex pressure amplitude ψ at any point (r,θ,φ) is known to satisfy the Helmholtz equation in a source free domain∇2ψ(x, k)+k2ψ(x, k)=0.  (2)
Outside a surface S that contains all acoustic sources in the scene, the potential ψ(x,k) is regular and satisfies the Sommerfeld radiation condition at infinity:
                                                                        lim                ⁢                                                                  ⁢                r                            ⁢                                                                                  r              →              ∞                                ⁢                      (                                                            ∂                  ψ                                                  ∂                  r                                            -                              ⅈ                ⁢                                                                  ⁢                k                ⁢                                                                  ⁢                ψ                                      )                          =        0                            (        3        )            
Outside S, the regular potential ψ(x,k) that satisfies equation (2) and condition (3) may be expanded in terms of singular elementary solutions (called multipoles). A multipole Φlm(x,k) is characterized by two indices m and l which are called order and degree, respectively. In spherical coordinates, x=(r,θ,φ)Φlm(r,θ,φ,k)=hl(kr)Ylm(θ,φ),   (4)Where hl (kr) are the spherical Hankel functions of the first kind, and Ylm(θ,φ) are the spherical harmonics,
                                          Y                          l              ⁢                                                          ⁢              m                                ⁡                      (                          θ              ,              φ                        )                          =                                            (                              -                1                            )                        m                    ⁢                                                                      (                                                            2                      ⁢                      n                                        +                    1                                    )                                ⁢                                  (                                      l                    -                                                                                          m                                                                    !                                                        )                                                            4                ⁢                                                                  ⁢                                  π                  ⁡                                      (                                          l                      +                                                                                                  m                                                                          !                                                              )                                                                                ⁢                                    P              l                                              m                                                      ⁡                          (                              cos                ⁢                                                                  ⁢                θ                            )                                ⁢                      ⅇ                          ⅈ              ⁢                                                          ⁢              m              ⁢                                                          ⁢              φ                                                          (        5        )            where Pn|m|(λ) are the associated Legendre functions.
In the arrangement, shown in FIG. 1, a representation of the potential in the region between the head and the many speaker locations is sought. Unfortunately this region contains sources (the speaker), and the scatterer, and thus does not satisfy the conditions for a fitting by multipoles (i.e., source free, and extending to infinity.
Therefore it would be highly desirable to provide a technique for rapid measurement of range dependent individualized HRTFs, correct interpolation procedures associated therewith, and procedures which permit development of HRTFs in terms of a series of multipole solutions of the Helmholtz equation.